Linear Transformation

Definition

A linear transformation is a mapping $L: V \to W$ between two vector spaces that preserves the operations of addition and scalar multiplication:

1. $L(u + v) = L(u) + L(v)$
2. $L(cu) = cL(u)$

• How to read: “The transformation L of u plus v equals L of u plus L of v, and L of c u equals c times L of u.”
• Meaning: $L$ respects vector addition and scaling—no translation, no curvature; grids stay evenly spaced.

Why It Matters

Mastering linear transformations is the key to manipulating multidimensional data; without this foundation, one cannot understand the internal engines of computer graphics, robotics, or the neural networks that define modern AI.

Core Concepts

• Matrix Representation: Every linear transformation between finite-dimensional spaces can be represented by a matrix $A$ such that $L(v) = Av$.

  • How to read: “The transformation L of v equals A times v.”
  • Meaning / when to use: Columns of $A$ are where basis vectors land—matrix multiplication computes $L(v)$.

• Action on Basis: To find the matrix $A$, apply $L$ to each basis vector of $V$ and write the results as columns of $A$.

• Linearity Properties:

  • $L(0) = 0$.
  • The transformation of a linear combination is the linear combination of the transformations.

• Geometry: Transformations include rotations, reflections, scaling, and projections. Translations are not linear unless represented in higher-dimensional homogeneous coordinates.

Connected Concepts

• Singular Value Decomposition
• Change Of Basis
• Matrix Algebra: Inverses